Does the Hearing Aid correct the music it sends, or does it also boost the sound from the outside (e.g. in "transparency" mode)?

The Hearing Aid feature boosts sound from the outside in Transparency mode. Media Assist corrects for all audio sent to the headphones.

Two separate features that complement each other very well.

That's pretty amazing, given the price. Hearing aids are typically a lot more expensive (probably also because they need to be medically approved, something like that?).